Video transcript

Let's do a couple of problems
graphing linear equations. They are a bunch of ways to
graph linear equations. What we'll do in this video
is the most basic way. Where we will just plot a
bunch of values and then connect the dots. I think you'll see
what I'm saying. So here I have an equation,
a linear equation. I'll rewrite it just in case
that was too small. y is equal to 2x plus 7. I want to graph this
linear equation. Before I even take out the graph
paper, what I could do is set up a table. Where I pick a bunch of x values
and then I can figure out what y value would
correspond to each of those x values. So for example, if x is equal
to-- let me start really low-- if x is equal to minus 2--
or negative 2, I should say-- what is y? Well, you substitute
negative 2 up here. It would be 2 times
negative 2 plus 7. This is negative 4 plus 7. This is equal to 3. If x is equal to-- I'm just
picking x values at random that might be indicative of--
I'll probably do three or four points here. So what happens when
x is equal to 0? Then y is going to be equal
to 2 times 0 plus 7. Is going to be equal to 7. I just happen to be
going up by 2. You could be going up
by 1 or you could be picking numbers at random. When x is equal to
2, what is y? It'll be 2 times 2 plus 7. So 4 plus 7 is equal to 11. I could keep plotting
points if I like. We should already have
enough to graph it. Actually to plot any line, you
actually only need two points. So we already have one
more than necessary. Actually, let me just do one
more just to show you that this really is a line. So what happens when
x is equal to 4? Actually, just to not go up by
2, let's do x is equal to 8. Just to pick a random number. Then y is going to be 2 times 8
plus 7, which is-- well this might go off of our graph
paper-- but 2 times 8 is 16 plus 7 is equal to 23. Now let's graph it. Let me do my y-axis
right there. That is my y-axis. Let me do my x-axis. I have a lot of positive values
here, so a lot of space on the positive y-side. That is my x-axis. And then I use the points x
is equal to negative 2. That's negative 1. That's 0, 1, 2, 3,
4, 5, 6, 7, 8. Those are our x values. Then we can go up
into the y-axis. I'll do it at a slightly
different scale because these numbers get large
very quickly. So maybe I'll do it in
increments of 2. So this could be 2, 4,
6, 8, 10, 12, 14, 16. I could just keep going
up there, but let's plot these points. So the first coordinate I have
is x is equal to negative 2, y is equal to 3. So I can write my coordinate. It's going to be the point
negative 2, 3. x is negative 2. y is 3. 3 would land right over there. So that's our first one,
negative 2, 3. Then our next point. 0, 7. We do it in that color. 0, 7. x is 0. Y is 7. Right there. 0, 7. We have this one
in green here. Point 2, 11. 2, 11 would be right
about there. And then this last point-- this
is actually going to fall off of my graph. 8, 23. That's going to be way
up here someplace. If you can even see
what I'm doing. This is 8, 23. If we connect the dots, you'll
see a line forms. Let me connect these dots. I've obviously hand drawn it, so
it might not be a perfectly straight line. If you had a computer do it, it
would be a straight line. So you could keep picking x
values and figuring out the corresponding y values. In the situation y is a function
of our x values. If you kept plotting every
point, you'll get every line. If you picked every possible x
and plotted every one, you get every point on the line. Let's do another problem. At the airport, you can
change your money from dollars into Euros. The service costs $5. and for every additional dollar,
you get EUR 0.7. Make a table for this and plot
the function on a graph. Use your graph to determine how
many Euros you would get if you give the office $50. I will write Euros is equal to--
so let's see, it's going to be dollars. So you're going to have
to give your dollars. Right off of the bat, they're
going to take $5. So dollars minus 5. So immediately this
service costs $5. And then everything that's
leftover-- this is your leftover-- you get EUR 0.7 for
every leftover dollars. You get 0.7 for whatever's
leftover. So this is the relationship. Now we can plot points-- we
could actually answer their question right off the bat. If you give them $50, we don't
even have to look at a graph. But we will look at a graph
right after this. So if you did Euros is equal
to-- if you have given them $50-- it would be 0.7
times 50 minus 5. You gave them 50. They took 5 as a service fee. So this is just $45 It would
be 0.7 times 45. I could do that right here. 45 times 0.7. 7 times 5 is 35. 4 times 7 is 28 plus 3 is 31. And then we have only one number
behind the decimal, only this 7. So it's 31.5. So if you give them $50, you're
going to get EUR 31.5. Euros, not dollars. So we answered their question,
but let's actually do it graphically. Let's do a table. Maybe I'll get a
calculator out. I'll refer to that
in a little bit. So let's say dollars
you give them. And how many Euros do you get? I'll just put a bunch
of random numbers. If you give them $5, they're
just going to take your $5 for the fee. You're going to get $5 minus
5, which is 0 times 0.7. So you're going to
get nothing back. So there's really no good reason
for you to do that. Then if you give them $10. What's going to happen? If you give them $10, 10
minus 5 is 5 times 0.7. You're going to get $3-- or
I should say EUR 3.50. 3.5 Euros, you'll get. Now what happens if
you give them $30? Actually let me say 25. If you give him $25,
25 minus 5 is 20. 20 times 0.7 is $14. I'll do one more value. Let's say you gave them $55. This makes the math easy
because then you subtract that 5 out. 55 minus 5 is 50 times
0.7 is $35. Is that right? Yep, that's right. You'll get EUR 35
I should say. These are all Euros. I keep wanting to say dollars. Let's plot this. All of these values are
positive, so I only have to draw the first quadrant here. And so the dollars-- let's go
in increments of 5, 10, 15, 20, 25, 30, 35, 40,
45, 50, 55. I made my x-axis a little
shorter than I needed to. All the way up to 55. And then the y-axis. I'll go in increments of 5. So that's 5, 10, 15,
20, 25, 30, 35. Well that's a little bit too
much of an increment. 35. Now let's plot these points. I give them $5. I get EUR 0. This right here is Euros. This is the dollars. The dollars is the independent
variable and we figure out the Euros from it. Or the Euros I get
is dependent on the dollars I get. If I give $10, I get EUR 3.50. 3.50-- it's hard to read. Maybe 3.50 would be right
around there. If I give $25, I get EUR 14. 25, 14 is right about there. Obviously, I'm hand drawing it,
so it's not going to be quite exact. If I get $55, I get EUR 35. So 55, 35 right there. If I were to connect to the
dots, I should get something that looks pretty
close to a line. If I did it-- if I was
a computer, it would be exactly a line. That looks pretty good. Then we could eyeball what
they asked us to do. Use your graph to determine how
many Euros you would get if you give the office $50. This is 50 right here. So you go bam, bam, bam, bam,
bam, bam, bam, bam. I'm at the graph. Then you go all the way--
actually I drew that last point on the graph a little
bit incorrectly. Let me. 35 is right here. Let me redraw that point. 35 is right there roughly. So 55, 35 is right there. So let me redraw my line. It will look-- I lost 25. 25, 14 is right there. So my graph looks something
like that. That's my best attempt. Now let's answer the question. We give them $50 right there. You go up, up, up, up,
up, up, up. $50. The person is going to get. You go all the way to
the left-hand side. That's right about 31.50. We figured out exactly
using the formula. But you can see, you can eyeball
it from the graph and figure out any amount
of dollars. If you give them $20, you're
going to go all the way over here. You'll figure out that it should
be-- well $20 should be about 7.50. The imprecision in my graph-- in
my drawing the graph makes it a little bit less exact. When you say 20 minus 5 is 15. 15 times-- actually it'll
be a little over $10, which is right. It's right over there. If you put $20 in there,
20 minus 5 is 15. 15 times 0.7 is $10.50,
which is right there. So you can look at any point
in the graph and figure out how many Euros you'll get. Let's do this one where
we'll do a little bit of reading a graph. The graph-- I think it said
use the graph below. Oh, the graph below shows
a conversion chart for converting between weight
in kilograms and weight in pounds. Use it to convert the following
measurements. We have kilograms here
and pounds here. So they want 4 kilograms into
weight into pounds. So if we look at this
right here, 4 kilograms is right there. We just follow where
the graph is. So 4 kilograms into pounds, it
looks like, I don't know, a little bit under 9 pounds. So a little bit less than--
so almost, I'll write almost 9 pounds. You can't exactly see. It's a little less than
9 pounds right there. 4 kilograms. Now 9 kilograms.
We go over here. 9 kilograms. Go all
the way up. That looks like almost
exactly 20 pounds. Here they say 12 pounds into
weight in kilograms. Actually kilograms is mass, but I
won't get particular. So 12 pounds. Go over here. Pounds. 12 pounds in kilograms
looks like 5 1/2. Approximately 5 1/2. And then 17 pounds
to kilograms. So 17 is right there. 17 pounds to kilograms looks
right about 7 1/2 kilograms. Anyway, hopefully that these
examples made you a little bit more comfortable with graphing
equations and reading graphs of equations. I'll see you in the
next video.